water
Article
Agent Based Modelling for Water Resource Allocation
in the Transboundary Nile River
Ning Ding 1 , Rasool Erfani 2 , Hamid Mokhtar 3 and Tohid Erfani 1, *
1
2
3
*
Department of Civil, Environmental and Geomatic Engineering, University College
London, London WC1E 6BT, UK; [email protected]
Department of Mechanical Engineering, Manchester Metropolitan University, Manchester M15 6BH, UK;
[email protected]
Department of Mathematics and Statistics, The University of Melbourne, Victoria 3010, Australia;
[email protected]
Correspondence: [email protected]; Tel.: +44-207-679-0673
Academic Editor: Ashantha Goonetilleke
Received: 10 December 2015 ; Accepted: 1 April 2016 ; Published: 7 April 2016
Abstract: Water resource allocation is the process of assessing and determining a mechanism on how
water should be distributed among different regions, sectors and users. Over the recent decades, the
optimal solution for water resource allocation has been explored both in centralised and decentralised
mechanisms. Conventional approaches are under central planner suggesting a solution which
maximises total welfare to the users. Moving towards the decentralised modelling, the techniques
consider individuals as if they act selfishly in their own favour. While central planner provides an
efficient solution, it may not be acceptable for some selfish agents. The contrary is true as well in
decentralised solution, where the solution lacks efficiency leading to an inefficient usage of provided
resources. This paper develops a parallel evolutionary search algorithm to introduce a mechanism
in re-distributing the central planner revenue value among the competing agents based on their
contribution to the central solution. The result maintains the efficiency and is used as an incentive for
calculating a fair revenue for each agent. The framework is demonstrated and discussed to allocate
water resources along the Nile river basin, where there exist eleven competing users represented
as agents in various sectors with upstream-downstream relationships and different water demands
and availability.
Keywords: agent based problems; water resource allocation; Nile River; evolutionary algorithm
1. Introduction
Water scarcity, population growth and lack of proper resource allocation mechanisms tend to cause
regional instability [1]. A typical example concerns the northern African countries within Nile basin
located in the most arid region of the world, where an unfair distribution of water resources has been
present for a long time. Introducing a fair mechanism for water allocation can help the region’s economy
and political stability. So far, the centralised system by a central planner (CP) has been a standard
water management approach, by which the whole water basin is modelled as a centralised system
and then water is distributed for maximising the total benefit of users. Centralised system techniques
assume that all agents will allocate the water among each other such that their aggregate welfare is
maximised [2–4]. In this mechanism, the water is allocated to achieve the equal marginal return for all
the users. This leads to an ambiguous interpretation of the aggregated problem. In [5], it is argued that
the aggregated problem (i.e., CP) carries strong institutional assumptions, presupposing either central
planning or perfectly functioning of the water market. In fact, the aggregated formulation (a) does
not recognise the asymmetric accessibility of the water to users (e.g., from upstream to downstream);
Water 2016, 8, 139; doi:10.3390/w8040139
www.mdpi.com/journal/water
Water 2016, 8, 139
2 of 13
(b) ignores the selfishness of competing water users; and (c) assumes the best solution to the system
would be accepted completely by all the participants. Therefore, the standard aggregated approach
(i.e., CP) is not practical when it is used to deal with sharing the water resource. To overcome the
above issues, decentralised planning (DC) is introduced. In [6] a priority based sequential algorithm
for upstream-downstream water reallocation is implemented. Once the upstream user maximises its
benefit, its decision (solution) is imposed to its immediate downstream user as predefined status; this
continues until all the individual problems are solved in sequence. The applicability of multi-agent
systems have also been investigated in the field of environmental and natural resource management as
reported in [7,8]. In this type of approach, each user is autonomous by itself and exchange information
with its neighbour users within a system. An example of using a multi-agent system is developed in [9],
and is further extended in allocation of water in the Yellow river basin [10] and is used to compare
administrative and market based water allocation [11]. This approach considers all users as individual
agents making decisions by interacting with each other and a coordinator who resolves the users’
conflict in later stages. The method implements the modified penalty-based nonlinear programme with
a two-step problem. The first step finds a solution to agents individually with a possibility of constraint
infeasibility and the second step is an optimisation model which reduces the constraint violation at
the system level. In application, constraint infeasibility is explained as either the deficit or as an agent
behavioural adjustment indicator for reducing the constraint violation [9]. From a game theoretical
perspective, non-cooperative approaches have been examined in the systems in which users involve in a
game to increase their pay-off, knowing that their decisions affect those of the other users. The approach
provides insights for understanding water conflicts and is often implemented for the games with
qualitative information about the users’ payoffs [12]. Another approach to the above problems is
developed in [5]. They use the multiple complementarity problems to express spatial externalities
resulting from asymmetric access to water use for water right pricing. The individual optimisation
problem is formulated for each user with the inflow quantity given as exogenous value to each
problem as opposed to being a decision variable in the centralised formulation, i.e., aggregated welfare
maximisation. The price of the demanded water is used to clear the output market and the uniform
wage rate is used to clear the labour market formulated as complementary constraints to the problem.
To this framework, introducing extra coupling constraints changes the formulation to a more general
problem framework namely, quasi variational inequality problem (i.e., a complementarity problem
with shared constraints amongst the users [13]). The convergence of the algorithm is guaranteed
upon the convexity assumption and continuously differentiable functions with diagonally dominant
Jacobians [14].
Fair Resource Allocation
Although the above decentralised tools and techniques satisfy the selfishness of each agent in
maximising its utility function to achieve higher revenue, they lead to an inefficient solution from
CP perspective. That is, it is possible within a water basin that the most inefficient agents located
upstream use water up to their operating capacity, and leave very limited units of water to the most
efficient firms located downstream; a situation explained in [5,15]. Therefore, it is desirable to allocate
the water based on the efficient CP solution, but to re-distribute the achieved revenue to the agents in
a fair way—considering, of course, that the revenue is transferable between agents. Different allocation
approaches in the literature considers different ways to address the fairness [16]. The distance based
methods, namely, least square solution, maximin (minimax) and compromise programming are some
mathematical methods which generally evaluate the performance of solutions based on their distance
from ideal solution. These are some reliable indicators to be used to quantify the dissatisfaction level
of a user within a shared system. To account for fairness, in this paper, a notion of fairness based on
each agent’s contribution on achieving the CP solution is defined. A unique solution is calculated with
some favourable properties which guarantees the cooperation maintenance. To find the agent’s impact
on CP solution, as will be discussed in the next sections, the best response of each agent on the action
Water 2016, 8, 139
3 of 13
of the other group of agents and vice versa should be known, simultaneously. Therefore, as a major
contribution of this paper, an evolutionary algorithm is developed solving interrelated optimisation
problems in parallel guiding the search towards a feasible solution in a distributed manner so that
the impact of agents on CP solution is realised. This will guarantee that the contribution of each
agent is properly captured for fair revenue distribution considering the conflicts within such a shared
resource system.
2. Problem Identification: Nile River Basin
Over the recent decades, decrease in exploitable freshwater resources and poorly performing
water resources management policies have escalated the water competition among river basin countries.
The accessibility to water becomes a growing concern in many water basins, especially in Middle
East and North Africa, where the water scarcity problems are almost the most severe around the
world [16]. In this region, the Nile basin, one of the most important water basins, is ensuring the
basic livelihood of people living along the river. It is the main vital water artery and the home to
more than 160 million people in the North Eastern region of Africa shared by eleven countries [17].
The Nile is 6853 kilometres in length and total area of its basin is over 3 million kilometres, covering
about 10 percent of the African continent [18]. There are two main tributaries: the White Nile and
the Blue Nile, which are joined in the Sudan (Figure 1). The White Nile originates in the Kagera
River in Burundi, passing through Lake Victoria and Lake Kyoga successively. The Blue Nile that
consists of numerous tributaries starts from Ethiopia, flowing northwards and merging the White
Nile into a single River. Rainfall is characterised by a highly uneven spatial distribution over the
basin. The reliability and volume of precipitation generally declines moving northwards, with the
arid regions in Egypt and the northern region of the Sudan receiving insignificant annual rainfall.
Therefore, the water contribution to the river varies greatly; from Ethiopia, which contributes the
most water, to Egypt, which has no contribution to Nile water [19]. However, as the lower reaches of
the Nile basin are mostly arid or semiarid regions, some countries such as Egypt and Sudan show a
strong dependency upon the Nile River [20] (Table 1). The unbalance between the insignificant water
availability and excessive water extraction cause harmful consequences to basin stability and regional
development. Hence, an adequate water supply is often considered as a question of national survival
for many Nile riparian states [21]. In addition, water use in the Nile basin causes a potential source of
conflict that overshadows many other types of collaboration.
Table 1. Utilisation of water diverted from the Nile River among riparian countries [22,23].
Country
Internal
Water
Resources
(IRWR)
Actual
Water
Resources
(ARWR)
Dependancy
Ratio
Diverted Water
from Nile
% of Total
Resources
Diverted
for Use
Burundi
Rwanda
Tanzania
Uganda
Sudan
S.Sudan
Egypt
Ethiopia
Eritrea
Congo
Kenya
10.06
9.5
84
39
4.0
26.0
1.8
122
2.8
900
20.7
12.54
13.3
96.27
60.1
37.8
49.5
58.3
122
7.315
1283
30.7
19.75
28.57
12.75
35.11
96.13
65.8
96.91
0
61.72
29.85
32.57
40.9
17.1
N/A
11.4
1074
1074
990
76
124.0
6.7
74.85
2.3
1.58
N/A
0.46
58
58
94.7
4.56
N/A
N/A
8.91
1.77
1.07
N/A
0.18
56
56
103
4.27
N/A
N/A
7.05
Water 2016, 8, 139
4 of 13
Figure 1. Nile River basin, its location and tributaries.
The allocation of Nile water resource is complicated due to the combination of riparian’s less
rainfall and political inequality. As the most powerful countries, Egypt and Sudan hold absolute
rights to use 100 percent of the river’s water under agreements reached in 1929 between Egypt and
Britain and in 1959 between Egypt and Sudan [18]. This apportions most of the water to Egypt and the
former Sudan, which makes other riparian states hardly meet their water demand, leading to series
of conflicts and issues about the water resource negotiation. With the changes of regional policies,
competition and conflicts over water development and utilisation between downstream and upstream
countries are increasingly aggravated since 1980s [17]. Specifically, the upstream countries represented
by Ethiopia are strongly disappointed with the condition that the large amount of water is extracted
by the two downstream countries, Sudan and Egypt. Since the late 1990s, the Nile riparian countries
initiated Nile River basin management cooperation, which launched a series of basin wide dialogue
and relevant joint activities, mobilised the process of negotiating and singing of the Nile River Basin
Cooperative Framework Agreement [20]. However, the political and economic factors as well as
dependency to water for downstream countries hinder the negotiating progress. The dependency to
water resources shown in Table 1 is the degree to which the supply of a country’s water resources
is dependent on sources external to its political boundaries and can be calculated using the relation
( ARWR − IRWR)/ARWR × 100 [17]. As shown in Table 1, Sudan and Egypt rely on the external
water resources to a great extent, in which over 95% of water stems from external sources. Overall, the
water allocation within the basin is still unfair and unacceptable to many of states along the Nile River,
specially to those upstream contributing the most to the sources.
3. Preliminaries and Definitions for Fair Resource Allocation
In this study, a fair and an efficient resource allocation approach based on evolutionary algorithm
(EA) is proposed. To retain the efficient centralised solution whilst the achieved revenue is fairly
re-distributed among the agents, the impact each agent has on the whole system should be identified.
In order to know the best response of each agent on the coalition of others, a parallel evolutionary
algorithm is developed by [15,24], enabling the agents to solve their local optimisation problem while
interacting with the others. To elaborate some key concepts mathematically, the preliminaries are
as follows.
Water 2016, 8, 139
5 of 13
3.1. Preliminary and Definitions
Let I = {1 . . . n} denotes a set of agents. Assume that each agent i controls vector xi ∈ Rni . Let x−i
be a vector containing the strategies (allocation) of all agents excluding that of the agent i. Each agent
by receiving allocation xi maximises his revenue via its utility function ui . The utility ui of the strategy
profile x = ( x1 , . . . , xn ) ∈ Rn+ or in short x = ( xi , x−i ) is ui (x) = ui ( xi , x−i ). The followings are defined.
Definition: (Central Planner Welfare Maximisation (CP))
A solution is a social welfare maximisation or a central planner (CP) approach if it is derived by
the following optimisation problem,
x∗ = argmax ∑ ui (x),
x
(CP)
i∈ I
where summation is over all the utilities of the agents. This leads to a solution from an outside observer
as if he/she is responsible for the values of all agents.
Definition: (Contribution to Cooperation)
Define U ∗ = ∑ j∈ I u j (x∗ ). Further, assume that agent i decides to leave the cooperation and act as
∗ =
∗ ) be the summation of all other agent’s revenue
a singleton (or in isolation) and let U−
∑ j 6 =i u j ( x −
i
i
when i leaves them. Agent i’s impact on CP solution is defined as,
∗
ui = U ∗ − U−
i.
ui measures how much agent i contributes to CP solution. In other words, ui is the impact of agent i
leaving the cooperation.
Definition: (Fairness)
A revenue re-distribution mechanism is fair if the revenue for each agent i follows the
following equation:
uri = αi × U ∗ ,
where,
αi =
ui
∗ .
∑ j U−
j
This means that each agent gets an allocation based on his contribution to the CP solution.
This definition makes sense and has two indirect properties; (a) it is budget balanced; that is, the sum
of all uri equals the whole CP revenue value U ∗ , which in other words conveys that the mechanism
collects and disburses the same amount of money from and to the agents; and, (b) it is rational; that is,
no agent ever loses by participation (the revenue to each user is greater than zero). The above explains
that the more contribution one agent has, the higher its revenue is. In this case, agents are encouraged
to abide by the decision derived by CP problem (x∗ ) if they are given a revenue following uri values.
∗ implies that agent i, which left the set of all agents, independently compete on the resources
U−
i
with agents {1, 2, . . . , i − 1, i + 1, . . . , n}. If agent i knew the others’ strategies, his strategic problem
would become simple; he would be left with the single-agent problem of choosing a utility-maximising
problem. However, the two problems formed by agent i and agents {1, 2, . . . , i − 1, i + 1, . . . , n} should
be solved, simultaneously. This is because of the fact that agent i’s best strategy depends on the
∗ values.
interaction with the group he has left and which should not be ignored when finding U−
i
∗
Therefore, U−i depends on the solution of two interrelated maximisation problems formed by agent
{i }’s utility, ui , and agents’ {1, 2, . . . , i − 1, i + 1, . . . , n} aggregated utilities, ∑ j6=i u j ( x−i ) which should
j∈ I
Water 2016, 8, 139
6 of 13
be solved at the same time. A parallel evolutionary technique is defined next to deal with this two
distributed problems.
3.2. Parallel Search Algorithm
Here, a general class of interrelated problems is formulated in which their optimisation problems
are simultaneously solved in parallel while interacting with each other. In a most general case and
where n agents are solving their problems individually, each agent solves one optimisation problem and
seeks its own optimal strategies while interacting with the others. More precisely, given U : Rn → Rn
representing all n agents’ utilities, x = ( x1 , . . . , xn ) ∈ Rn+ is found by simultaneously solving the
following n problems:
Max
xi
u i (x)
subject to x ∈ Xi ,
(Pi )
where each agent i controls vector xi ∈ Rni to optimise the utility (objective) function ui subject to the
constraints set Xi containing x ∈ Rn+ . The interrelation is explained as the objective function and the
constraints in Pi depend on other agents’ decisions.
To solve the n agent problems Pi , i = 1, ..., n simultaneously, each problem Pi is dedicated to
one agent i. Since there is interconnection between each problem due to vector x, each problem is
solved whilst it communicates with the other problems by sharing information. Call P the problem
formed by all Pi s. Parallel genetic algorithm [25] developed in [24] and the idea of co-evolution [26]
are implemented to solve P. The idea is extended such that each (sub-)problem Pi has its own objective
function. This concept is used in [27] to gain faster convergence to Pareto solution in multiobjective
optimisation problem. Let x−i be a vector containing the decision variables of all agents involved
in problem Pi excluding that of the agent i. The search algorithm is described by n different search
trajectories performing in parallel through the following mapping H:
xit+1 = H ( x t−i , xit , Pi ),
where H shows the interconnection between the agents. H acts as a synchronization map for agent i to
optimise problem Pi given the decisions of other interacting agents in its neighbourhood remain fixed
shown by x t−i . H describes that xi value is updated by a search on problem Pi at generation t linking
decisions xi and x −i . Due to problem Pi , each agent knows its own problem components and hence by
communicating with other neighbouring agents through H, it has local activity for exploring the search
space. In what follows, Algorithm 1 gives details of the search algorithm to solve the agents problems.
Each agent i has a devoted search trajectory formed by a population of size m (Line 1). popi is
a m × nei matrix and is populated randomly. nei is the number of interacting agents given by the
cardinal of the set neighboursi (Line 2). In other words, nei equals the number of neighbouring agents
affecting the decision of agent i plus one. All individuals pk = ( x1 , ..., xnei ) in each population i
undergoes a reproduction in each generation t of parallel searches (Line 8). pb is reproduced from two
other distinct individuals within the population. If its objective value is better than that of the pk , it
remains in the population otherwise it is discarded (Line 9 and 10). At the end of each generation t,
the neighbouring agents (j ∈ neighboursi ) share their best individuals to form the updated population
for next generation t + 1 (Line 12).
Figure 2 explains the algorithm where two agents are involved in the system. As explained in
the figure, each agent deals with problem Pi optimising for xi . At the end of each generation t, popi∗ ,
the best individual in popi based on its objective value, is obtained. popi∗ migrates to the population
of the neighbours and remain fixed for the next generation t + 1. This makes each agent at the end
of each generation to be informed of the decisions of the other neighbouring agents involved in its
own problem. Due to n different search trajectories, the algorithm allows independent search for
agents by relying only on locally available information. This procedure leads to the evolution of
Water 2016, 8, 139
7 of 13
separate populations over successive generations, and the convergence is assumed when the agents
cannot further improve their objective function values f i (readers may refer to ([24] Section 3.2) for an
illustrative example of Algorithm 1).
1: Algorithm 1: Parallel search algorithm
1
2
3
4
5
6
7
8
9
10
Randomly initialise n populations of size m (popi );
Define neighboursi and set neii = |neighboursi |;
Set MaxGen;
while Not MaxGen do
for i = 1 to n do
for k = 1 to m do
Corresponding to pk , randomly pick mutually distinct ps1 and ps2 from popi ;
pb ← reproduction (ps1 , ps2 );
if f i (pb ) ≤ f i (pk ) then
pk ← pb
popi∗ ← The best individual in popi ;
11
12
For any pair of (i, j) such that i 6= j and j ∈ neighboursi , let popi ← pop∗j ;
xi
xi
xi+1
xi+1
Agent
1
i+1
2
Agent
m
Problem Pi
xi
variable
xi+1 remain fixed
xi
Problem Pi+1
xi
remain fixed
xi+1 variable
i
xi+1
Figure 2. Exchanging the best individual values within the the neighbouring populations at the end of
each generation for nei = 2. xi∗ is fixed in popi+1 and xi∗+1 is fixed in popi in each generation.
3.3. Resource Allocation Context
As stated earlier, to find the contribution ui of each agent i to the CP solution, it should be
assured that the solution to agent i’s utility maximisation is the best response to the solution of sum
of utilities of the other agents and vice versa. To do so, the set I is split by removing one agent
at a time from I to form two problems P1 and P2 for each instances. Specifically, problem P1 is
the utility maximisation for agent i (ui ) and problem P2 is the aggregated utility maximisation for
Water 2016, 8, 139
8 of 13
agents 1, 2, . . . , i − 1, i + 1, . . . , n (∑ j6=i u j ( x−i )). Problem P1 and P2 are then solved in parallel for each
j∈ I
agent i using Algorithm 1. The illustrative procedure summarises the steps to obtain a fair resource
allocation to different self-interested agents. For each agent i, problem P1 and P2 are solved in parallel
in line 2 and 3. The contribution of each agent to the system then is calculated in line 4 and based on
the fairness definition, the revenue is re-distributed in line 6.
2: Illustrative procedure: Steps to redistribute utilities amongst self-interested agents
1
2
3
4
5
6
Find U ∗ ;
for i = 1 to n do
Solve problem P1 and P2 using Algorithm 1;
For each agent i, calculate ui , αi ;
uri ← αi × U ∗ ;
Distribute to each agent uri ;
4. Nile River Basin Water Sharing Mechanism
Considering the major water utilisation of riparian and their geographic positions (Figure 1), the
water users located in the Nile riparian states are modelled as agents within a distribution network.
In this paper, economic concepts is used to translate the demand for water to an economic value.
Water demands are usually represented in hydroeconomic models using exogenously generated linear
or quadratic equations relating water application to economic benefits [4]. In some cases, complex
crop yield functions are explicitly included in the model as well [28]. A detailed introduction of
the economics of water resources can be found in [29,30]. A water demand curve shows the user’s
willingness to pay for demanded water. The x- and y-axis are the water quantity available to be
abstracted and the unit price or marginal willingness to pay, respectively. The gross economic benefits
of water abstraction is calculated by integrating the demand curve [4]. While any complex economic
model could be exploited, without loss of generality, in the modelling presented in this paper, the
economic objective function is a quadratic function shown by ai xi − bi xi2 and calculated by integrating
the linear water demand functions for each agent [15] (for details, the reader is referred to [2,3,31]).
This function can be modified to closely represent the case study specific assumptions, for example,
to consider that the marginal utility in arid countries is not likely to reach zero. All agents follow the
upstream-downstream relationship, interconnecting with neighbours using the mass balance equations.
The CP model aims at the maximisation of total benefit, and is formulated as a single optimisation
problem with summation of all benefit functions as in Equation CP. Following Section 3.3, for the
decentralised model, agent i is separated from the rest of the agents and its own economic function is
maximised concurrently as the rest try to maximise their group revenue using Algorithm 1.
4.1. Water Availability
The mean annual flow of Nile River in 2015 is 84 billion cubic metre (BCM) per year [32]. In this
case-specific modelling, the minor water inflows and evaporative losses are not considered. If such
information exists, the inflows can be adjusted accordingly and a parameter can be included in the
model to account for water loss due to evaporation. Specific to the two tributaries, hydrological data
at Mogren dam is chosen to represent average annual runoff of the White Nile (Q1 ) and Khartoum
monitors data of the Blue Nile (Q2 ) [22]. In experimental set-up, therefore, Q1 = 24.0 BCM and,
Q2 = 60.0 BCM based on the average hydrological data regulated at these stations [33].
4.2. Population and Demand Values
The objective function is the benefit function that quantifies the total benefit generated by water
extractors from water use. In order to set a reasonable value for the parameters ai and bi in an
Water 2016, 8, 139
9 of 13
objective function, the water demand curves should be estimated primarily according to the water
demand and price, and then total benefit functions are calculated by integrating the demand functions.
Following [2], the point expansion method is used to estimate the linear demand curve for various
sectors. The original point expansion is based on the total water consumption and the water price.
For simplicity, the marginal value of water is referenced as water price. Water consumption is
obtained using:
Water demand = total water usage × % o f population within the basin.
Table 2 exhibits the factors determining the total water demand in the basin amongst agents.
Table 2. Water consumption within the basin [20,22].
Agent
Sectors
BU
RW
TA
CO
UG
KE
SS
ET
ER
SU
EG
Agriculture
Agriculture
Agriculture
Agriculture
Industry
Agriculture
Energy
Agriculture
Agriculture
Agriculture
Municipal
Population Within
the Basin (Million)
% of Total
Population
Water Usage
(BCM)
Water Demand
with the Basin (BCM)
Source
4.88
8.17
8.24
2.8
30.28
14.62
10
29.56
0.21
20
51
44.50%
69.40%
16.70%
4.10%
76.40%
33.00%
85.50%
31.40%
3.30%
29.60%
62.20%
0.22
0.1
4.632
0.11
0.12
1.01
0.21
5.204
0.29
6.56
5.3
0.0979
0.0694
0.7749
0.0046
0.0917
0.3329
0.1818
1.6347
0.0096
1.9445
3.2941
1
1
1
1
1
1
1
2
2
1+2
1+2
The population within the basin, water usage for utilisation and their marginal values are the
main benchmarks when determining the water demand curves, which are indirectly reflected on
parameters setting in objective functions [17]. Based on Table 2, the values of ai and bi are tabulated in
Table 3.
Table 3. The parameters defining each agent’s economic objective function.
Agent
BU
RW
TA
CO
UG
KE
SS
ET
ER
SU
EG
a
b
100
511
100
721
100
65
100
10960
1860
10139
100
150
13000
35757
100
31
100
5200
100
26
1300
197
5. Results and Discussion
In both CP and decentralised solution procedure, MaxGen = 100 in Algorithm 1, population size
for each agent is set as m = 50 and cross over and mutation is set as 0.5 and 0.7 for reproduction,
respectively. Accounting for reliability, all the instances are run for 30 times and their average value
is reported.
5.1. Centralised Solution
In CP model, the fitness function is the aggregated benefit of all countries and, therefore, the
problem is to search the maximum value of system revenue . The revenue of the whole system is
reported as 3575.94 B. The benefits of each agent i in CP solution are shown in Table 4 along with the
amount of water abstracted.
Water 2016, 8, 139
10 of 13
Table 4. Water resource allocation results in centralised manner (CP). Burundi(BU), Rwanda(RW),
Tanzania(TA), Congo(CO), Uganda(UG), Kenya(KE), S.Sudan(SS), Ethiopia(ET), Eritrea(ER),
Sudan(SU), Egypt(EG).
Agent
BU
RW
TA
CO
Water (bcm)
Benefit (mGBP)
Total benefit
0.1
4.9
0.04
2.8
0.54
35
0
0
UG
KE
SS
ET
0.08 0.17
0.16
1.24
84.7 12.5 1159.5 76.3
U ∗ = ∑ ui∗ = 3575.94
ER
SU
EG
0
0
1.72
95.1
2.85
2105.1
5.2. Decentralised Solution
Eleven different model instances are solved where in each single instance, two problems are
optimised in parallel using Algorithm 1. Table 5 reports the results.
Table 5. Water resource allocation results.
Agent
Country
Contribution
ūi
Singleton
P1
Group
P2
Fairness
αi
Final
Revenue uri
BU
RW
TA
CO
UG
KE
SS
ET
ER
SU
EG
Burundi
Rwanda
Tanzania
Congo
Uganda
Kenya
S.Sudan
Ethiopia
Eritrea
Sudan
Egypt
76.94
75.94
219.94
86.94
157.94
81.94
1226.94
139.94
45.94
168.94
2216.94
4.89
2.802
35
0
85.09
13.01
1168
76.37
0
96.01
1947
3499
3500
3356
3489
3418
3494
2349
3436
3530
3407
1359
0.017
0.017
0.049
0.019
0.035
0.018
0.273
0.031
0.01
0.038
0.493
61.16
60.37
174.84
69.11
125.55
65.14
975.35
111.24
36.52
134.3
1762.35
5.3. Re-Allocation Solution
After finding the decentralised solution, from the perspective of fairness, the system revenue is
reallocated based on the results derived from CP solution (Table 4). Figure 3 shows the contributions
of each agent. The difference between the CP value and the group value of the rest in decentralised
model embodies the impact one agent has on the whole system. Hence, the contribution is calculated,
which provides the basis for revenue re-distribution. The incentive of agents in a cooperation game
is determined by their location. The downstream users with high water dependency usually have
higher incentive to join the cooperation. Figure 4 compares the decentralised solution with the CP
distribution. For example, agent C contributes more than its upstream user B since it has less access
to the water resource yet it requires more water resources. It can be seen that upstream location is
beneficial to agents compared with the CP solution. Agent A, Burundi, who has the independent
water resource as the upstream of White Nile tributary (Q1 ), could increase its final obtainable benefit
greatly from 4.9 to 61.16 in million pounds. This is the same for the other upstream users, while, on the
contrary, the two main downstream water abstractors, agent G and K, are apportioned with less water
after re-distribution. Through the rearrangement of water allocation, the upstream-downstream water
disputes has the potential to be reduced. In addition, the distribution tends to be more evenly among
agents than that in CP solution, which could be explained as the reflection of fairness to some extent.
Water 2016, 8, 139
11 of 13
TANZANIA
5%
RWANDA
2%
CONGO
2%
BURUNDI
2%
UGANDA
4%
KENYA
2%
EGYPT
SOUTH
SUDAN
49%
27%
ERITREA
1%
ETHIOPIA
SUDAN
3%
4%
Figure 3. Percentages of contribution in cooperation.
BURUNDI
RWANDA
TANZANIA
CONGO
EGYPT
UGANDA
KENYA
SOUTH
SUDAN
ETHIOPIA
SUDAN
ERITREA
ETHIOPIA
ERITREA
SUDAN
SOUTH
SUDAN
EGYPT
KENYA
UGANDA
CONGO
TANZANIA
RWANDA
CP Solution
Re-Distributed Solution
Water Flow
BURUNDI
Figure 4. Revenue allocation results in central planner (CP) solution and reallocation solution.
6. Conclusions
This paper introduces a methodological framework to address the Nile River water allocation
problem through a revenue re-distribution mechanism. The proposed framework leads to a fairly
allocated revenue for each user, which is proportional to its contribution to the basin. In a centralised
solution, aggregated benefits of all water users are used to search the optimal system revenue and in a
decentralised solution, a parallel evolutionary approach is developed to find the contribution of each
user to the whole system. The evolutionary algorithm is a parallel search where each user solves his/her
own problem while in contact with the others. Re-allocation of revenue in this framework guarantees
a fair and an efficient allocation of water to all users. Geographical location of users as well as their
Water 2016, 8, 139
12 of 13
sector they are involved in (manifested via different marginal values) are the main factors affecting the
final available revenue for water users, which, in turn, determine their contributions. Compared with
the centralised solution, the results have taken into account the selfishness of individuals, providing a
fairer distribution of water to those with greater accessibility to the water. The revenue distribution
mechanism introduced in this paper is a fair and unique approach, but its stability requires further
investigation. In addition, the algorithmic characteristics of the proposed framework still needs
to be explored. Future research can analyse the technique for feasibility assurance and possibly
faster convergence by using different operators and heuristics. The limitation of the approach with
many agents within a system should also be explored. However, since n instances of problems are
independent from each other, a parallelisation scheme can be implemented to remedy the shortcomings.
Author Contributions: Ning Ding produced the first set of results and wrote part of the literature review and
background of Nile river. She also calculated the economical benefit functions and coded the Nile model.
Rasool Erfani wrote the algorithm section, produced the main idea of agent based modelling and wrote part of
the literature review. Hamid Mokhtar helped the coding of the algorithm, wrote part of the text and produced a
figure. Tohid Erfani conducted the research, wrote the main part of the algorithm text, coded the algorithm and
reviewed the manuscript with added references and background text.
Conflicts of Interest: The authors declare no conflict of interest.
References
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
Speed, R.; Yuanyuan, L.; Zhiwei, Z.; Le Quesne, T.; Pegram, G. Basin Water Allocation Planning: Principles,
Procedures and Approaches for Basin Allocation Planning; UNESCO: Paris, France, 2013.
Erfani, T.; Binions, O.; Harou, J.J. Simulating water markets with transaction costs. Water Resour. Res. 2014,
50, 4726–4745.
Erfani, T.; Binions, O.; Harou, J. Protecting environmental flows through enhanced water licensing and
water markets. Hydrol. Earth Syst. Sci. Discuss. 2014, 11, 2967–3003.
Harou, J.J.; Pulido-Velazquez, M.; Rosenberg, D.E.; Medellín-Azuara, J.; Lund, J.R.; Howitt, R.E.
Hydro-economic models: Concepts, design, applications, and future prospects. J. Hydrol. 2009, 375, 627–643.
Britz, W.; Ferris, M.; Kuhn, A. Modeling water allocating institutions based on Multiple Optimization
Problems with Equilibrium Constraints. Environ. Model. Softw. 2013, 46, 196–207.
Wang, L.; Fang, L.; Hipel, K.W. Basin-wide cooperative water resources allocation. Eur. J. Oper. Res. 2008,
190, 798–817.
Barreteau, O.; Bousquet, F.; Millier, C.; Weber, J. Suitability of Multi-Agent Simulations to study irrigated
system viability: Application to case studies in the Senegal River Valley. Agric. Syst. 2004, 80, 255–275.
Bousquet, F.; Barreteau, O.; d’Aquino, P.; Etienne, M.; Boissau, S.; Aubert, S.; Page, C.L.; Babin, D.;
Castella, J.C.; Janssen, M.; et al. Multi-agent systems and role games: Collective learning processes for
ecosystem management. In Complexity and Ecosystem Management: The Theory and Practice of Multi-Agent
Systems; ACM Digital Library: New York, NY, USA, 2002; pp. 248–285.
Yang, Y.C.E.; Cai, X.; Stipanović, D.M. A decentralized optimization algorithm for multiagent system–based
watershed management. Water Resour. Res. 2009, 45, doi:10.1029/2008WR007634.
Yang, Y.C.E.; Zhao, J.; Cai, X. Decentralized Optimization Method for Water Allocation Management in the
Yellow River Basin. J. Water Resour. Plan. Manag. 2011, 138, 313–325.
Zhao, J.; Cai, X.; Wang, Z. Comparing administered and market-based water allocation systems through a
consistent agent-based modeling framework. J. Environ. Manag. 2013, 123, 120–130.
Madani, K. Game theory and water resources. J. Hydrol. 2010, 381, 225–238.
Heusinger, A.V. Numerical Methods for the Solution of the Generalized Nash Equilibrium Problem.
Ph.D. Thesis, Wuĺrzburg University, Würzburg, Germany, 2009.
Kolstad, C.D.; Mathiesen, L. Computing Cournot-Nash equilibria. Oper. Res. 1991, 39, 739–748.
Erfani, T.; Erfani, R. Fair Resource Allocation Using Multi-population Evolutionary Algorithm. In Applications
of Evolutionary Computation; Springer: Berlin, Germany, 2015; pp. 214–224.
Read, L.; Madani, K.; Inanloo, B. Optimality versus stability in water resource allocation. J. Environ. Manag.
2014, 133, 343–354.
Water 2016, 8, 139
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
13 of 13
El-Fadel, M.; El-Sayegh, Y.; El-Fadl, K.; Khorbotly, D. The Nile River Basin: A case study in surface water
conflict resolution. J. Natl. Resour. Life Sci. Educ. 2003, 32, 107–117.
Kameri-Mbote, P.G. Water, Conflict, and Cooperation: Lessons from the Nile River Basin; Woodrow Wilson
International Center for Scholars: Washington, DC, USA, 2007.
Mohamoda, D.Y. Nile Basin Cooperation: A Review of the Literature; Number 26; Nordic Africa Institute:
Uppsala, Sweden, 2003.
Hu, W.; Yang, J.; Huang, H. A study of water resources development and utilization and management
cooperation across the Nile River Basin. Resour. Sci. 2011, 10, 003.
Hamouda, M.A.; El-Din, M.M.N.; Moursy, F.I. Vulnerability assessment of water resources systems in the
Eastern Nile Basin. Water Resour. Manag. 2009, 23, 2697–2725.
Food and Agriculture Organization of the United Nations Regional Office for Africa. FAO Statistical Yearbook
2014 Africa Food and Agriculture; FAO Uited Nation: Accra, Ghana, 2014.
Food and Agriculture Organization of the United Nations. Aquastat FAO; FAO Uited Nation: Accra, Ghana,
2015.
Erfani, T.; Erfani, R. An evolutionary approach to solve a system of multiple interrelated agent problems.
Appl. Soft Comput. 2015, 37, 40–47.
Mühlenbein, H.; Schomisch, M.; Born, J. The parallel genetic algorithm as function optimizer. Parallel Comput.
1991, 17, 619–632.
Potter, M.A.; De Jong, K.A. Cooperative coevolution: An architecture for evolving coadapted subcomponents.
Evolut. Comput. 2000, 8, 1–29.
Lee, D.; Gonzalez, L.F.; Periaux, J.; Srinivas, K. Efficient hybrid-game strategies coupled to evolutionary
algorithms for robust multidisciplinary design optimization in aerospace engineering. IEEE Trans.
Evolut. Comput. 2011, 15, 133–150.
Cai, X.; McKinney, D.C.; Rosegrant, M.W. Sustainability analysis for irrigation water management in the
Aral Sea region. Agric. Syst. 2003, 76, 1043–1066.
Gibbons, D.C. The Economic Value of Water; Routledge: New York, NY, USA, 2013.
Griffin, R.C. Water Resource Economics: The Analysis of Scarcity, Policies, and Projects; MIT Press Books:
Cambridge, MA, USA, 2006; Volume 1.
Erfani, T.; Huskova, I.; Harou, J.J. Tracking trade transactions in water resource systems: A node-arc
optimization formulation. Water Resour. Res. 2013, 49, 3038–3043.
Understanding Nile Basin Hydrology: Mapping Actual Evapotranspiration Over The Nile Basin. In Technical
Bulletin from the Nile Basin Initiative Secretariat; Nile Waters: Entebbe, Uganda, 2015.
Sutcliffe, J.V.; Parks, Y.P. The Hydrology of the Nile; Number 5; International Association of Hydrological
Sciences Wallingford: Oxfordshire, UK, 1999.
c 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access
article distributed under the terms and conditions of the Creative Commons by Attribution
(CC-BY) license (http://creativecommons.org/licenses/by/4.0/).